Let A be a set of real numbers
and F be a class of complex-valued functions defined on the real line such that for
each f ∈ F the infinite series S(x,f) =∑k=1∞f(kαj) converges for every nonzero x
in A. If 0 ∈ A, we set S(0,f) = f(0). It seems to be an interesting problem to study
the different sets A and function classes F such that each f ∈ F is uniquely
determined by the sums S(x,f) where x ∈ A. Clearly, the larger the class F is
studied, the larger set A is needed to guarantee uniqueness. We have positive results
for a class of entire functions of exponential type and for fairly large classes of
continuous functions. Some examples are also given to show that in general A cannot
be too small.
